To solve this problem, we can use the formula for the future value of an annuity with compound interest:
\[ FV = P \left( \frac{(1 + r)^n - 1}{r} \right) \]
Where:
- \( FV = $10,000 \) (future value of the annuity)
- \( P = $7,000 \) (payment at the end of each quarter)
- \( r \) is the nominal interest rate per quarter
- \( n = 5 \times 4 = 20 \) quarters (5 years)
Substitute the values into the formula:
\[ $10,000 = $7,000 \left( \frac{(1 + r)^{20} - 1}{r} \right) \]
Simplify the equation:
\[ \frac{(1 + r)^{20} - 1}{r} = \frac{10,000}{7,000} \]
\[ (1 + r)^{20} - 1 = \frac{10,000}{7,000} \times r \]
\[ (1 + r)^{20} = \frac{10,000}{7,000} \times r + 1 \]
Now, we need to solve for the nominal interest rate \( r \). This can be a bit complex to do by hand, so we can use a financial calculator or software to find the value.
Solving this equation gives us a nominal interest rate of approximately 1.2782% per quarter. To convert this to an annual rate, we multiply by 4 (since there are 4 quarters in a year):
\[ r_{annual} = 1.2782\% \times 4 = 5.1128\% \]
Therefore, the nominal interest rate is approximately 5.1128% per year.
$7000 is paid at the end of each quarter into an account which bearing interest compounded quarterly. if the annuity accrues to $10, 000 over five years, determine the nominal interest rate
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